Subgroups — #10765747

Order 768.

598 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#78475	C₂	1	1	yes	maximal, characteristic
2	#78445	C₂	1	2	yes	maximal
3	C₂×C₃×#322	C₂	1	1	yes	maximal, characteristic
4	#69738	C₂	1	2	yes	maximal
5	#69730	C₂	1	1	yes	maximal, characteristic
6	#68089	C₂	1	4	yes	maximal, complemented
7	#68084	C₂	1	4	yes	maximal, complemented
8	#17961	—	3	1	no	maximal, Sylow
9	#5470	C₂²	1	2	yes	—
10	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	#5116	C₄	1	4	yes	—
14	#5102	—	2	2	no	—
15	#5101	—	2	2	no	—
16	#4924	C₂²	1	2	yes	—
17	#4699	C₄	1	1	yes	characteristic
18	C₄×Dic₁₂	C₂²	1	4	yes	—
19	#4682	C₄	1	1	yes	characteristic
20	#4681	C₄	1	2	yes	—
21	#4679	—	2	2	no	—
22	#4679	C₂²	1	4	yes	—
23	C₈×Dic₆	—	2	2	no	—
24	C₈×Dic₆	C₂²	1	4	yes	—
25	C₃×#322	C₂²	1	2	yes	—
26	C₃×#322	C₂²	1	2	yes	—
27	#4457	C₂²	1	4	yes	—
28	#4456	C₂²	1	8	yes	—
29	#3170	—	3	1	no	—
30	#2960	—	3	2	no	—
31	#1593	—	3	4	no	—
32	#1579	—	3	4	no	—
33	#1566	—	3	1	no	—
34	#1493	—	3	2	no	—
35	C₂×#322	D₃	1	1	yes	characteristic
36	C₂²×C₃×C₈	—	2	1	no	—
37	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
38	C₂²×C₃×C₈	D₄	1	2	yes	—
39	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
40	#926	—	2	2	no	—
41	#924	—	2	2	no	—
42	#924	C₂×C₄	1	4	yes	—
43	C₂×#208	—	2	2	no	—
44	C₂×Dic₁₂	C₂×C₄	1	4	yes	—
45	#870	—	2	2	no	—
46	#869	C₂×C₄	1	2	yes	—
47	#869	C₂×C₄	1	2	yes	—
48	C₄×Dic₆	—	2	2	no	—
49	C₄×Dic₆	C₂³	1	4	yes	—
50	C₃×C₄×C₈	—	2	1	no	—
51	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
52	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
53	C₃×C₄×C₈	D₄	1	2	yes	—
54	C₃×C₄×C₈	C₂³	1	2	yes	—
55	C₃×C₄×C₈	C₂³	1	2	yes	—
56	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
57	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
58	#818	C₂×C₄	1	4	yes	—
59	#816	C₂×C₄	1	8	yes	—
60	#815	—	2	4	no	—
61	Dic₃⋊C₈	—	2	4	no	—
62	Dic₃⋊C₈	—	2	4	no	—
63	C₈×Dic₃	—	2	4	no	—
64	C₄⋊(C₃⋊C₈)	—	2	2	no	—
65	C₄⋊(C₃⋊C₈)	C₂³	1	4	yes	—
66	#533	—	3	2	no	—
67	#500	—	3	1	no	—
68	#494	—	3	1	no	—
69	#487	—	3	2	no	—
70	#467	—	3	4	no	—
71	#446	—	3	4	no	—
72	#446	—	6	2	no	—
73	C₈×Q₈	—	3	4	no	—
74	C₈×Q₈	—	6	2	no	—
75	C₄×Q₁₆	—	3	4	no	—
76	#424	—	6	2	no	—
77	#423	—	3	2	no	—
78	#405	—	6	2	no	—
79	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
80	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
81	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
82	#334	—	3	4	no	—
83	Q₈⋊C₈	—	3	8	no	—
84	#322	C₂×D₃	1	2	yes	—
85	#322	C₂×D₃	1	2	yes	—
86	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₃×C₄	#41	1	1	yes	characteristic
89	C₂×Dic₆	—	2	2	no	—
90	C₂×Dic₆	C₂²×C₄	1	4	yes	—
91	C₂×C₃×C₈	—	2	4	no	—
92	C₂×C₃×C₈	—	2	4	no	—
93	C₂×C₃×C₈	—	2	2	no	—
94	C₂×C₃×C₈	—	4	2	no	—
95	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
97	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
98	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
99	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
100	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
101	C₃×C₄²	#41	1	1	yes	characteristic
102	C₃×C₄²	C₂⁴	1	1	yes	characteristic
103	#217	—	2	4	no	—
104	C₄⋊Dic₃	—	2	2	no	—
105	C₄⋊Dic₃	—	2	2	no	—
106	C₄⋊Dic₃	—	2	2	no	—
107	C₄⋊Dic₃	C₂²×C₄	1	4	yes	—
108	Dic₃⋊C₄	—	2	4	no	—
109	Dic₃⋊C₄	—	2	4	no	—
110	C₄×Dic₃	—	2	4	no	—
111	#208	—	4	4	no	—
112	C₂×C₃⋊C₈	—	2	4	no	—
113	Dic₁₂	—	2	8	no	—
114	C₂×Q₁₆	—	3	4	no	—
115	C₂×M₁₆	—	6	2	no	—
116	C₂²×C₈	—	2	1	no	—
117	C₂²×C₈	C₄×D₃	1	2	yes	—
118	C₂²×C₈	D₁₂	1	2	yes	—
119	C₄⋊Q₈	—	3	2	no	—
120	C₄⋊Q₈	—	3	2	no	—
121	#125	—	6	2	no	—
122	C₂²⋊Q₈	—	3	4	no	—
123	C₂²⋊Q₈	—	6	2	no	—
124	C₄×Q₈	—	3	4	no	—
125	C₄×Q₈	—	6	2	no	—
126	#117	—	6	2	no	—
127	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
128	#107	—	3	4	no	—
129	C₄⋊C₈	—	3	4	no	—
130	C₄⋊C₈	—	6	4	no	—
131	C₄⋊C₈	—	6	4	no	—
132	C₄⋊C₈	—	6	2	no	—
133	#103	—	3	8	no	—
134	#97	—	6	4	no	—
135	C₄×C₈	—	2	1	no	—
136	C₄×C₈	—	6	4	no	—
137	C₄×C₈	C₄×D₃	1	1	yes	characteristic
138	C₄×C₈	C₄×D₃	1	1	yes	characteristic
139	C₄×C₈	D₁₂	1	2	yes	—
140	C₄×C₈	C₂²×D₃	1	2	yes	—
141	C₄×C₈	C₂²×D₃	1	2	yes	—
142	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
143	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
144	C₂³×C₃	C₄×D₄	1	1	yes	characteristic
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	2	no	—
147	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
148	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
149	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
150	C₂×C₃×C₄	#137	1	4	yes	—
151	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
154	C₂×Dic₃	—	2	4	no	—
155	C₂×Dic₃	—	2	4	no	—
156	Dic₆	—	2	8	no	—
157	Dic₆	—	4	4	no	—
158	C₃×C₈	—	2	2	no	—
159	C₃×C₈	—	2	2	no	—
160	C₃×C₈	—	2	2	no	—
161	C₃×C₈	—	4	2	no	—
162	C₃×C₈	#131	1	4	yes	—
163	C₃⋊C₈	—	4	4	no	—
164	C₂×Q₈	—	3	4	no	—
165	C₂×Q₈	—	6	2	no	—
166	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
168	C₂²×C₄	#235	1	1	yes	characteristic
169	Q₁₆	—	6	8	no	—
170	M₁₆	—	12	4	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	6	4	no	—
176	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
177	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
178	C₂×C₈	C₂×D₁₂	1	2	yes	—
179	C₂×C₈	C₂×D₁₂	1	2	yes	—
180	C₄⋊C₄	—	3	4	no	—
181	C₄⋊C₄	—	6	2	no	—
182	C₄⋊C₄	—	6	2	no	—
183	C₄⋊C₄	—	6	4	no	—
184	C₄⋊C₄	—	6	2	no	—
185	C₄⋊C₄	—	6	4	no	—
186	C₂²⋊C₄	—	6	4	no	—
187	C₄²	—	6	4	no	—
188	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
189	C₄²	C₂×D₁₂	1	1	yes	characteristic
190	C₄²	#235	1	1	yes	characteristic
191	C₄²	C₂³×D₃	1	1	yes	characteristic
192	C₂²×C₃	—	2	1	no	—
193	C₂²×C₃	—	2	1	no	—
194	C₂²×C₃	#441	1	2	yes	—
195	C₂²×C₃	C₂×C₄×D₄	1	1	yes	characteristic
196	C₃×C₄	—	2	1	no	—
197	C₃×C₄	—	2	1	no	—
198	C₃×C₄	#441	1	2	yes	—
199	C₃×C₄	#444	1	2	yes	—
200	C₃×C₄	C₂×#131	1	2	yes	—
201	C₃×C₄	C₂×#137	1	2	yes	—
202	Dic₃	—	4	4	no	—
203	Dic₃	—	4	4	no	—
204	C₂³	C₄×D₁₂	1	1	yes	characteristic
205	Q₈	—	6	8	no	—
206	Q₈	—	12	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	6	4	no	—
210	C₂×C₄	—	6	4	no	—
211	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
212	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
213	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
214	C₂×C₄	#909	1	4	yes	—
215	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
216	C₂×C₄	C₂²×D₁₂	1	1	yes	characteristic
217	C₂×C₄	C₂×#235	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	2	2	no	—
220	C₈	—	2	2	no	—
221	C₈	—	4	2	no	—
222	C₈	—	12	4	no	—
223	C₈	#901	1	4	yes	—
224	C₂×C₃	—	2	2	no	—
225	C₂×C₃	#2931	1	1	yes	characteristic
226	C₂×C₃	C₂×#441	1	1	yes	characteristic
227	C₂×C₃	C₂×#444	1	1	yes	characteristic
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	#4716	1	2	yes	—
231	C₂²	C₂×C₄×D₁₂	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	12	4	no	—
235	C₄	—	12	4	no	—
236	C₄	#4716	1	2	yes	—
237	C₄	#4717	1	2	yes	—
238	C₄	C₂×#901	1	2	yes	—
239	C₄	C₂×#909	1	2	yes	—
240	C₃	#17961	1	1	yes	characteristic, complemented, Sylow
241	C₂	—	2	2	no	—
242	C₂	#78458	1	1	yes	characteristic
243	C₂	C₂×#4716	1	1	yes	characteristic
244	C₂	C₂×#4717	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₂
commutator	C₂×C₃×C₄	C₂³×C₄
Frattini	C₄²	C₂³×D₃
Fitting	C₂×C₃×#322	C₂
socle	C₂²×C₃	C₂×C₄×D₄
prime radical	C₂³×C₃	C₄×D₄
cosocle	C₃×C₄²	C₂⁴
hypercenter	C₂×#322	D₃
hypocenter	C₃	#17961
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
6. #68089	C₂	semidirect
7. #68084	C₂	semidirect
240. C₃	#17961	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#78475	2
#78445	2
C₂×C₃×#322	2
#69738	2
#69730	2
#68089	2
#68084	2
#17961	3